NCERT Exemplar solutions for Mathematics [English] Class 12 chapter 10 - Vector Algebra [Latest edition]

Find the unit vector in the direction of the sum of the vectors `vec"a" = 2hat"i" - hat"j" + 2hat"k"` and `vec"b" = -hat"i" + hat"j" + 3hat"k"`.

Find a vector of magnitude 11 in the direction opposite to that of `vec"PQ"` where P and Q are the points (1, 3, 2) and (–1, 0, 8), respectively.

Find the position vector of a point R which divides the line joining the two points P and Q with position vectors `vec"OP" = 2vec"a" + vec"b"` and `vec"OQ" = vec"a" - 2vec"b"`, respectively, in the ratio 1:2 internally

Find the position vector of a point R which divides the line joining the two points P and Q with position vectors `vec"OP" = 2vec"a" + vec"b"` and `vec"OQ" = vec"a" - 2vec"b"`, respectively, in the ratio 1:2 externally

If the points (–1, –1, 2), (2, m, 5) and (3,11, 6) are collinear, find the value of m.

Find a vector `vec"r"` of magnitude `3sqrt(2)` units which makes an angle of `pi/4` and `pi/2` with y and z-axes, respectively.

If `vec"a" = 2hat"i" - hat"j" + hat"k", vec"b" = hat"i" + hat"j" - 2hat"k"` and `vec"c" = hat"i" + 3hat"j" - hat"k"`, find `lambda` such that `vec"a"` is perpendicular to `lambdavec"b" + vec"c"`.

Find all vectors of magnitude `10sqrt(3)` that are perpendicular to the plane of `hat"i" + 2hat"j" + hat"k"` and `-hat"i" + 3hat"j" + 4hat"k"`

Using vectors, prove that cos (A – B) = cosA cosB + sinA sinB.

Prove that in a ∆ABC,  `sin"A"/"a" = sin"B"/"b" = sin"C"/"c"`, where a, b, c represent the magnitudes of the sides opposite to vertices A, B, C, respectively.

The magnitude of the vector `6hat"i" + 2hat"j" + 3hat"k"` is ______.

The position vector of the point which divides the join of points with position vectors `vec"a" + vec"b"` and 2`vec"a" - vec"b"` in the ratio 1:2 is ______.

The vector with initial point P (2, –3, 5) and terminal point Q(3, –4, 7) is ______.

The angle between the vectors `hat"i" - hat"j"` and `hat"j" - hat"k"` is ______.

The value of λ for which the two vectors `2hati - hatj + 2hatk` and `3hati + λhatj + hatk` are perpendicular is ______.

The area of the parallelogram whose adjacent sides are `hat"i" + hat"k"` and `2hat"i" + hat"j" + hat"k"` is ______.

If `|vec"a"|` = 8, `|vec"b"|` = 3 and `|vec"a" xx vec"b"|` = 12, then value of `vec"a" * vec"b"` is ______.

The 2 vectors `hat"j" + hat"k"` and `3hat"i" - hat"j" + 4hat"k"` represents the two sides AB and AC, respectively of a ∆ABC. The length of the median through A is ______.

The projection of vector `vec"a" = 2hat"i" - hat"j" + hat"k"` along `vec"b" = hat"i" + 2hat"j" + 2hat"k"` is ______.

If `veca` and `vecb` are unit vectors, then what is the angle between `veca` and `vecb` for `sqrt(3)  veca - vecb` to be a unit vector?

The unit vector perpendicular to the vectors `hat"i" - hat"j"` and `hat"i" + hat"j"` forming a right handed system is ______.

If `|vec"a"|` = 3 and –1 ≤ k ≤ 2, then `|"k"vec"a"|` lies in the interval ______.

Find the unit vector in the direction of the sum of the vectors `vec"a" = 2hat"i" - hat"j" + hat"k"` and `vec"b" = 2hat"j" + hat"k"`.

If `vec"a" = hat"i" + hat"j" + 2hat"k"` and `vec"b" = 2hat"i" + hat"j" - 2hat"k"`, find the unit vector in the direction of `6vec"b"`

If `vec"a" = hat"i" + hat"j" + 2hat"k"` and `hat"b" = 2hat"i" + hat"j" - 2hat"k"`, find the unit vector in the direction of `2vec"a" - vec"b"`

Find a unit vector in the direction of `vec"PQ"`, where P and Q have co-ordinates (5, 0, 8) and (3, 3, 2), respectively

If `vec"a"` and `vec"b"` are the position vectors of A and B, respectively, find the position vector of a point C in BA produced such that BC = 1.5 BA.

Using vectors, find the value of k such that the points (k, – 10, 3), (1, –1, 3) and (3, 5, 3) are collinear.

A vector `vec"r"` is inclined at equal angles to the three axes. If the magnitude of `vec"r"` is `2sqrt(3)` units, find `vec"r"`.

A vector `vec"r"` has magnitude 14 and direction ratios 2, 3, – 6. Find the direction cosines and components of `vec"r"`, given that `vec"r"` makes an acute angle with x-axis.

Find a vector of magnitude 6, which is perpendicular to both the vectors `2hat"i" - hat"j" + 2hat"k"` and `4hat"i" - hat"j" + 3hat"k"`.

Find the angle between the vectors `2hat"i" - hat"j" + hat"k"` and `3hat"i" + 4hat"j" - hat"k"`.

If `vec"a" + vec"b" + vec"c"` = 0, show that `vec"a" xx vec"b" = vec"b" xx vec"c" = vec"c" xx vec"a"`. Interpret the result geometrically?

Find the sine of the angle between the vectors `vec"a" = 3hat"i" + hat"j" + 2hat"k"` and `vec"b" = 2hat"i" - 2hat"j" + 4hat"k"`.

If A, B, C, D are the points with position vectors `hat"i" + hat"j" - hat"k", 2hat"i" - hat"j" + 3hat"k", 2hat"i" - 3hat"k", 3hat"i" - 2hat"j" + hat"k"`, respectively, find the projection of `vec"AB"` along `vec"CD"`.

Using vectors, find the area of the triangle ABC with vertices A(1, 2, 3), B(2, – 1, 4) and C(4, 5, – 1).

Using vectors, prove that the parallelogram on the same base and between the same parallels are equal in area.

Prove that in any triangle ABC, cos A = `("b"^2 + "c"^2 - "a"^2)/(2"bc")`, where a, b, c are the magnitudes of the sides opposite to the vertices A, B, C, respectively.

If `vec"a", vec"b", vec"c"` determine the vertices of a triangle, show that `1/2[vec"b" xx vec"c" + vec"c" xx vec"a" + vec"a" xx vec"b"]` gives the vector area of the triangle. Hence deduce the condition that the three points `vec"a", vec"b", vec"c"` are collinear. Also find the unit vector normal to the plane of the triangle.

Show that area of the parallelogram whose diagonals are given by `vec"a"` and `vec"b"` is `(|vec"a" xx vec"b"|)/2`. Also find the area of the parallelogram whose diagonals are `2hat"i" - hat"j" + hat"k"` and `hat"i" + 3hat"j" - hat"k"`.

If `vec"a" = hat"i" + hat"j" + hat"k"` and `vec"b" = hat"j" - hat"k"`, find a vector `vec"c"` such that `vec"a" xx vec"c" = vec"b"` and `vec"a"*vec"c"` = 3.

The vector in the direction of the vector `hat"i" - 2hat"j" + 2hat"k"` that has magnitude 9 is ______.

The position vector of the point which divides the join of points `2vec"a" - 3vec"b"` and `vec"a" + vec"b"` in the ratio 3:1 is ______.

The vector having initial and terminal points as (2, 5, 0) and (–3, 7, 4), respectively is ______.

The angle between two vectors `vec"a"` and `vec"b"` with magnitudes `sqrt(3)` and 4, respectively, and `vec"a" * vec"b" = 2sqrt(3)` is ______.

Find the value of λ such that the vectors `vec"a" = 2hat"i" + lambdahat"j" + hat"k"` and `vec"b" = hat"i" + 2hat"j" + 3hat"k"` are orthogonal ______.

The value of λ for which the vectors `3hat"i" - 6hat"j" + hat"k"` and `2hat"i" - 4hat"j" + lambdahat"k"` are parallel is ______.

The vectors from origin to the points A and B are `vec"a" = 2hat"i" - 3hat"j" + 2hat"k"` and `vec"b" = 2hat"i" + 3hat"j" + hat"k"`, respectively, then the area of triangle OAB is ______.

For any vector `vec"a"`, the value of `(vec"a" xx hat"i")^2 + (vec"a" xx hat"j")^2 + (vec"a" xx hat"k")^2` is equal to ______.

If `|vec"a"|` = 10, `|vec"b"|` = 2 and `vec"a".vec"b"` = 12, then value of `|vec"a" xx vec"b"|` is ______.

The vectors `lambdahat"i" + hat"j" + 2hat"k", hat"i" + lambdahat"j" - hat"k"` and `2hat"i" - hat"j" + lambdahat"k"` are coplanar if ______.

If `vec"a", vec"b", vec"c"` are unit vectors such that `vec"a" + vec"b" + vec"c"` = 0, then the value of `vec"a" * vec"b" + vec"b" * vec"c" + vec"c" * vec"a"` is ______.

Projection vector of `vec"a"` on `vec"b"` is ______.

If `vec"a", vec"b", vec"c"` are three vectors such that `vec"a" + vec"b" + vec"a" = vec0` and `|vec"a"|` = 2, `|vec"b"|` = 3, `|vec"c"|` = 5, then value of `vec"a"*vec"b" + vec"b"*vec"c" + vec"c"*vec"a"` is ______.

If `|vec"a"|` = 4 and −3 ≤ λ ≤ 2, then the range of `|lambdavec"a"|` is ______.

The number of vectors of unit length perpendicular to the vectors `vec"a" = 2hat"i" + hat"j" + 2hat"k"` and `vec"b" = hat"j" + hat"k"` is ______.

The vector `vec"a" + vec"b"` bisects the angle between the non-collinear vectors `vec"a"` and `vec"b"` if ______.

If `vec"r" * vec"a" = 0, vec"r" * vec"b" = 0` and `vec"r" * vec"c" = 0` for some non-zero vector `vec"r"`, then the value of `vec"a" * (vec"b" xx vec"c")` is ______.

The vectors `vec"a" = 3hat"i" - 2hat"j" + 2hat"k"` and `vec"b" = -hat"i" - 2hat"k"` are the adjacent sides of a parallelogram. The acute angle between its diagonals is ______.

The values of k for which `|"k"vec"a"| < |vec"a"|` and `"k"vec"a" + 1/2 vec"a"` is parallel to `vec"a"` holds true are ______.

The value of the expression `|vec"a" xx vec"b"|^2 + (vec"a".vec"b")^2` is ______.

If `|vec"a" xx vec"b"|^2 + |vec"a".vec"b"|^2` = 144 and `|vec"a"|` = 4, then `|vec"b"|` is equal to ______.

If `vec"a"` is any non-zero vector, then `(vec"a" .hat"i")hat"i" + (vec"a".hat"j")hat"j" + (vec"a".hat"k")hat"k"` equals ______.

If `|vec"a"| = |vec"b"|`, then necessarily it implies `vec"a" = +- vec"b"`.

Position vector of a point P is a vector whose initial point is origin.

If `|vec"a" + vec"b"| = |vec"a" - vec"b"|`, then the vectors `vec"a"` and `vec"b"` are orthogonal.

The formula `(vec"a" + vec"b")^2 = vec"a"^2 + vec"b"^2 + 2vec"a" xx vec"b"` is valid for non-zero vectors `vec"a"` and `vec"b"`

If `vec"a"` and `vec"b"` are adjacent sides of a rhombus, then `vec"a" * vec"b"` = 0